Make the dough: Sprinkle the yeast over the hot water in a small bowl. Let stand until dissolved and slightly foamy, 5 to 10 minutes.
Meanwhile, whisk the bread flour, all-purpose flour, sugar and salt in a large bowl and make a well in the center. Pour the yeast mixture into the well and add the olive oil. Stir with a wooden spoon until a shaggy dough forms.
Turn the dough out onto a lightly floured surface, dust the dough with flour and knead, dusting with more flour as needed, until very smooth and elastic but still slightly tacky, 3 to 5 minutes.
Transfer the dough to an oiled bowl, turning to coat. Cover with plastic wrap and let rise in a warm place until doubled in size, about 1 1/2 hours. Divide the dough into two 1-pound balls; wrap one ball in plastic and refrigerate for up to 2 days or freeze for up to 3 months. Bring to room temperature before using.
Make the pizza sauce: Combine the tomatoes, garlic, olive oil, dried oregano, sugar and a pinch each of salt and pepper in a medium bowl; season with more salt and sugar if needed. (Makes about 1 1/2 cups.)
Make the pizza: Preheat the oven to 500˚ F (use the convection setting, if available) with a pizza stone, baking steel or inverted large baking sheet in the lower third of the oven. Working on a lightly floured or cornmeal-dusted piece of parchment, gently stretch and press the other ball of dough into a 12- to 13-inch round. Cover the dough with 1/2 cup pizza sauce, the Parmesan, torn mozzarella and grated mozzarella, leaving a 3/4-inch border.
Transfer the pizza (on the parchment) to the hot stone and bake until the crust is set, about 5 minutes. Slide out the parchment from under the pizza; continue to bake until the crust is golden brown and the cheese is melted, 5 to 10 minutes. Slide the pizza onto a cutting board.